The novel you are looking for is Variable Star by Spider Robinson, based on notes that Robert A. Heinlein made for a novel. The basic story is:
When Joel Johnston first met Jinny Hamilton, it seemed like a dream come true. And when she finally agreed to marry him, he felt like the luckiest man in the universe.
There was just one small problem. He was broke. His only goal in life was to become a composer, and he knew it would take years before he was earning enough to support a family.
But Jinny wasn't willing to wait. And when Joel asked her what they were going to do for money, she gave him a most unexpected answer. She told him that her name wasn't really Jinny Hamilton---it was Jinny Conrad, and she was the granddaughter of Richard Conrad, the wealthiest man in the solar system.
And now that she was sure that Joel loved her for herself, not for her wealth, she revealed her family's plans for him--he would be groomed for a place in the vast Conrad empire and sire a dynasty to carry on the family business.
Most men would have jumped at the opportunity. But Joel Johnston wasn't most men. To Jinny's surprise, and even his own, he turned down her generous offer and then set off on the mother of all benders. And woke up on a colony ship heading out into space, torn between regret over his rash decision and his determination to forget Jinny and make a life for himself among the stars.
He was on his way to succeeding when his plans--and the plans of billions of others--were shattered by a cosmic cataclysm so devastating it would take all of humanity's strength and ingenuity just to survive.
Joel is indeed a jazz saxophone player. In fact, the reason he is able to afford his passage on a colony ship is his ability to play sax. Entertainment is at a premium during the twenty-year voyage. Musicians who agree to play during the voyage are welcomed with open arms.
Joel had a sort of inheritance from his father, who had invested in a ship that was sent to found a colony. The ship was never heard from, so Joel's father's investments were thought to be worthless. During the voyage, it was discovered that the ship had found a colony but their telepathic twin had died, so there was no communication until the child of that twin grew up and learned he was able to communicate with his uncle.
When the communication was re-established, Joel's apparently worthless investments were worth millions. He was convinced to use his inheritance to invest in the ship he was on.
The sun does indeed explode about halfway through the book.
However, they do not get confirmation of this until the first FLT ship, built by the scientist who eventually marries Jinny Conrad, is tested. The test flight is successful and takes place about a half-hour before the sun novas, saving Richard Conrad, Jinny, Jinny's scientist/husband, her younger sister, and the pilot. They need supplies, so they stop at the ship Joel is on.
The pilot and Conrad suggest that they could pay for supplies by ferrying the colonists to their new planet using the FTL drive ship. That would be good but they'd have to leave all their supplies and arrive at their colony planet with any supplies. Joel asks the inventor if the FTL drive could be recalibrated or installed in their ship.
At the end, Joel and his new wife, Jinny's younger sister, are using the first FLT ship to warn all the other colony planets about the Sun's nova radiation arrival and their need to be a few miles underground to survive and set up communications between the colony planets. Communication between colonies had been facilitated by Earth, which was (obviously) no longer possible.